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Let a model of the simply typed lambda calculus be a Cartesian-closed functor from $C_T$ to Set, where $C_T$ is a free CCC (as in e.g. this reference, p. 83.) The simple case of one or two primitive types with no constants or equations is good enough for me.

I'm interested in finding a pair of models $M, N : C_T → \text{Set}$ and a natural transformation $i : M → N$ such that each function $i_X : M(X) → N(X)$ is injective, but not every $i_X$ is surjective. Do such natural transformations exist?

Jeff Russell
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  • I still don't know the general answer to this question, but one thing that I think we have established is that for any such $i : M → N$, if $i_X$ is injective but not surjective, then $M(X)$ must be infinite. This follows from the answer to this question, together with the fact that for each $k$, the $k$th-power function is in the image of $M$. – Jeff Russell Mar 29 '17 at 22:02

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